Collatz Sequence Proof (2nd Way)

Abstract of Collatz Sequence Solution (2nd Way) I discovered a fact in Collatz Sequence: S (n) = {a, b, c,…,w} ⇒ LS(n) = LS(a) = LS(b) = LS(c) = …. = LS (w) … (Fact1) n, a, b, c, w, r ∈ N_+, LS(n) = {4,2,1} when n ∈ {1,2,3,4,5, …, r-1} Let LS(r) = {4,2,1} when n ∈ Z = {1,2,3,4,5, …, r-1, r} Part a) If (r+1) ∈ (N_even), and I proved LS(r+1) = {4,2,1} ⇒ LS(n) = {4,2,1} for all even #s Part b) If (r+1) ∈ (N_odd), and I proved LS(r+1) = {4,2,1} by part a. Therefore LS(N) = {4,2,1} for all n ∈N.